Derivational derivatives and Feynman's operational calculi

This paper explores the differential (or derivational) calculus associated with the disentangled operators arising from Feynman's operational calculi (FOCi) for noncommuting operators. (We will use the continuous case of the approach to FOCi initiated by Jefferies and Johnson in 2000.) The central part of this paper deals with a first order infinitesimal calculus for a function of n noncommuting operators. Here the first derivatives (or differentials) are replaced by the first order derivational derivatives. The derivational derivatives of the first and higher order have been useful in, for example, operator algebras, noncommutative geometry and Maslov's discrete form of Feynman's operational calculus. In the last section of this paper we will develop some special cases of higher order expansions.